[[Linear code]]
# Linear equivalence of codes

Let $\mathcal{C}, \mathcal{D} \leq \mathbb{K}_{q}^n$ be [[Linear code|linear codes]].
Then $\mathcal{C}, \mathcal{D}$ are said to be **linearly equivalent** iff there exists a [[monomial transformation]] $\varphi : \mathbb{K}_{q}^n \to \mathbb{K}_{q}^n$ such that $\varphi(\mathcal{C}) = \mathcal{D}$.[^2011] #m/def/code 
Equivalently, [[Linear code#^generator|generator matrices]] $G, G'$ of $\mathcal{C},\mathcal{D}$ respectively are related by permutation and rescaling of columns.

  [^2011]: 2011\. [[Sources/@liuEquivalenceLinearCodes2011|On the equivalence of linear codes]]

## Properties

- [[MacWilliams theorem]]

## See also

- [[Automorphism group of a linear code]]

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